How to delay all keyframes to make space for a new animation scene?
To solve your problem, you can select all (animated) objects in your scene, then select all keyframes that you want to move, and hit G to translate them further ahead.
To prevent this problem in the future, I suggest splitting your animation into shorter sequences, but this is case specific, not useable for when you want to extend the scene.
Since I don't know what exactly is going on there, I can only suggest.